f_0 <- function(t) 0.10 * t^2 / (0.7 + 0.04 * pmax(0, t - 3)^3)
g_0 <- function(t) 0.15 * t^2 / (0.9 + 0.01 * pmax(0, t - 1)^3)
f_1 <- function(t) 0.32 * exp(-0.15 * t)
f_until_1 <- function(t) 2.50 * exp(-0.60 * t)
g_1 <- function(t) 0.14 * exp(-0.25 * t)
g_until_1 <- function(t) 0.14 * exp(-0.25 * t)
f_0_1 <- f_0
f_0_3 <- g_0
f_1_2 <- function(t) 0.48 * exp(-0.10 * t)
f_1_3 <- function(t) 0.16 * exp(-0.30 * t)
formulas_dgp_timeScales <- list(
list(from = 0, to = 1,
formula = ~
f_0(tend) + beta_0_01 + beta_1_01 * x1 + beta_2_01 * x2
),
list(
from = 0, to = 3,
formula = ~
g_0(tend) + beta_0_03 + beta_1_03 * x1 + beta_2_03 * x2
),
list(
from = 1, to = 2,
formula = ~
f_0(tend) + f_1(t_1) + f_until_1(t_until_1) + beta_0_12 + beta_1_12 * x1 + beta_2_12 * x2
),
list(
from = 1, to = 3,
formula = ~
g_0(tend) + g_1(t_1) + g_until_1(t_until_1) + beta_0_13 + beta_1_13 * x1 + beta_2_13 * x2
)
)
formulas_dgp_stratified <- list(
list(from = 0, to = 1,
formula = ~
f_0_1(tend) + beta_0_01 + beta_1_01 * x1 + beta_2_01 * x2
),
list(
from = 0, to = 3,
formula = ~
f_0_3(tend) + beta_0_01 + beta_1_03 * x1 + beta_2_03 * x2
),
list(
from = 1, to = 2,
formula = ~
f_1_2(tend) + f_until_1(t_until_1) + beta_0_01 + beta_1_12 * x1 + beta_2_12 * x2
),
list(
from = 1, to = 3,
formula = ~
f_1_3(tend) + g_until_1(t_until_1) + beta_0_01 + beta_1_13 * x1 + beta_2_13 * x2
)
)Simulating Index Event Bias
Simulation Parameters
Distribution And Effect Size Of Risk Factors
Data Generating Processes
The effect sizes of the risk factors are similar to those of the UMOD SNP.
Other Parameters
cut <- seq(0, 10, by = 0.1)
terminal_states <- c(2, 3)
n <- 5000
round <- 1
cens_type <- "right"
cens_dist <- "weibull"
cens_params <- c(1.5, 10.0) # shape, scale
bs <- "ps"
k <- 20Model Formulas
formula_mod_timeScales <- ped_status ~
s(tend, by = trans_to_3, bs = bs, k = k) +
s(t_1, by = trans_after_1, bs = bs, k = k) +
s(t_until_1, by = trans_after_1, bs = bs, k = k) +
transition * x1 + transition * x2
formula_mod_timeScales_ieb <- ped_status ~
s(tend, by = trans_to_3, bs = bs, k = k) +
s(t_1, by = trans_after_1, bs = bs, k = k) +
s(t_until_1, by = trans_after_1, bs = bs, k = k) +
transition * x1
formula_mod_stratified <- ped_status ~
s(tend, by = transition, bs = bs, k = k) +
s(t_until_1, by = trans_after_1, bs = bs, k = k) +
transition * x1 + transition * x2
formula_mod_stratified_ieb <- ped_status ~
s(tend, by = transition, bs = bs, k = k) +
s(t_until_1, by = trans_after_1, bs = bs, k = k) +
transition * x1Correlations
Bernoulli-distributed omitted risk factor (p=0.5)
Bernoulli-distributed included risk factor (p=0.5)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=1)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=5)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed omitted risk factor (sd=1)
Bernoulli-distributed included risk factor (p=0.5)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=1)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=5)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed omitted risk factor (sd=5)
Bernoulli-distributed included risk factor (p=0.5)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=1)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=5)
Moderate Effect Sizes
Strong Effect Sizes
Conclusion
- No negative correlation in state 0 (by construction)
- No / little negative correlation in state 1 when omitted risk factor x2 is binary
- Negative correlation in state 1 when omitted risk factor x2 is normally distributed, increasing in size with increasing variance of x2, up to -0.40
Coefficients
Bernoulli-distributed omitted risk factor (p=0.5)
Bernoulli-distributed included risk factor (p=0.5)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=1)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=5)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed omitted risk factor (sd=1)
Bernoulli-distributed included risk factor (p=0.5)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=1)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=5)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed omitted risk factor (sd=5)
Bernoulli-distributed included risk factor (p=0.5)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=1)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=5)
Moderate Effect Sizes
Strong Effect Sizes
Conclusion
- No IEB when omitted risk factor x2 is binary
- IEB for both 0->1 and 1->2 when omitted risk factor x2 is normally distributed, increasing in size with increasing variance of x2, causing attenuation bias of up to 25% + 25% = 50%
Coverage
Bernoulli-distributed omitted risk factor (p=0.5)
Bernoulli-distributed included risk factor (p=0.5)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=1)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=5)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed omitted risk factor (sd=1)
Bernoulli-distributed included risk factor (p=0.5)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=1)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=5)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed omitted risk factor (sd=5)
Bernoulli-distributed included risk factor (p=0.5)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=1)
Moderate Effect Sizes
Strong Effect Sizes
Normally distributed included risk factor (sd=5)
Moderate Effect Sizes
Strong Effect Sizes
Conclusion
- Good coverage when omitted risk factor x2 is binary
- Coverage worsens significantly when omitted risk factor x2 is normally distributed, even though the bias (see coefficient plots) is sometimes very small